Compact open topology on $\mathrm{Homeo}(X)$

Question

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)\subset O$, where $K$ is any compact subset of $X$, and $O$ is any open subset of $Y$. So a basis of open sets is given by the following subsets: $[K_1,\dots,K_n,O_1,\dots,O_n]=[K_1,O_1 ]\cap\dots\cap [K_n,O_n]$, the collection of continuous maps $f:X\rightarrow Y$ that send each $K_i$ into $O_i$ for some specified collection of compact $K_i$'s and open $O_i$'s.

This topology has some nice properties: the exponential law holds under some hypotheses on the spaces $X$ and $Y$, and is certainly true if all spaces involved are locally compact Hausdorff spaces, as will be the case from now on.

My question is as follows: if $X$ is a locally compact Hausdorff space (or even a topological manifold), the compact open topology induces a topology on the set of homeomorphisms of $X$, which is a group. Does this topology turn $\mathrm{Homeo}(X)$ into a topological group? I can show that the product (composition) is continuous, but is the inverse too? $(f\rightarrow f^{-1})$

I was able to prove continuity for compact spaces, where it is very easy to establish. I also managed to prove it for $X=\mathbb{R}$ because all homeomorphisms of $\mathbb{R}$ are monotone, but that's everything so far.

I tried looking it up in several textbooks on topology and algebraic topology where the C.O. topology is usually discussed, but couldn't find a discussion on this topic anywhere.

Answer 1

The following article gives you a lot of information on the question you are asking:

On Homeomorphism Groups and the Compact-Open Topology, Jan J. Dijkstra

http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf

http://www.jstor.org/pss/30037630

The answer is in general "no".

Answer 2

For a simple counterexample, let X be the subspace of R consisting of 0 and exp(n) for all integers n, and consider the homeomorphisms $f_n$ of X defined by $f_n(0) = 0$, $f_n(\exp(k)) = \exp(k-1)$ for $k \le -n$ or $k > n$, $\exp(k)$ for $-n < k < n$, and $\exp(-n)$ for $k = n$. Then $f_n$ converges to the identity map in the compact-open topology, but $f_n^{-1}$ does not: $f_n^{-1}$ is not in the neighbourhood of the identity map given by $K = X \cap [0, 1]$ and $U = X \cap (-1, 2)$ for any $n \ge 1$, because $\exp(-n) \in K$ and $f_n^{-1}(\exp(-n)) = \exp(n) \notin U$.

Answer 3

R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946) 593–610.

If $X$ is locally compact and locally connected (!), then $\mathrm{Homeo}(X)$ is a topological group.

Answer 4

I don't know if this helps, but you could consider the set $[K,O]'$ of all maps $f\in Homeo(X)$ such that $f(K)\subset O$ and $f^{-1}(K)\subset O$. With the sets $[K,O]'$ as a subbase of neighborhoods of the identity, $Homeo(X)$ becomes a topological group, if $X$ is a locally compact $T_3$-space. This topology is called the Braconnier topology. It coincides with the compact open topology whenever $X$ is compact, or locally connected. In general it is stronger than the compact open topology.

Unfortunately I don't know any references for this facts, except the lecture notes of the 1982/83 lectures of Prof. Holdgruen on harmonic analysis (in a shelf in the library of the mathematics institute in Goettingen), because I learned this in my undergrad courses and not from books.